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        <h2 class="post-title" itemprop="name headline">模拟退火——史上最简单的智能算法

          
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        <p>一篇很久之前写的文章了，不过过了这么多年，模拟退火由于其效果和实现简单的优势，依然是智能算法中一个比较热门的算法，故老博客搬运而来。</p><h1 id="简介"><a href="#简介" class="headerlink" title="简介"></a>简介</h1><p>模拟退火算法从外部来看就是一个优化问题的解析器,我们给他传递初始解和产生新解的方法,它就能不断产生新解,并比较最终返回一个近似最优解.由于数学建模对算法的时间限制不严,而模拟退火又较易于实现,因此它也是数学建模里较常用的一种智能算法.<img src="/数模-模拟退火——史上最简单的智能算法/sa.jpg" alt="sa"></p><a id="more"></a>

<h1 id="快速使用"><a href="#快速使用" class="headerlink" title="快速使用"></a>快速使用</h1><p>在介绍具体算法前,我们完全可以在短时间内使用上模拟退火.</p>
<h2 id="例1-求min-x-2-y-2-x-y∈R"><a href="#例1-求min-x-2-y-2-x-y∈R" class="headerlink" title="例1:求min(x^2+y^2),x,y∈R."></a>例1:求<code>min(x^2+y^2),x,y∈R</code>.</h2><ul>
<li><p>首先,我们提供一个初始解.文件<code>main.m</code>.</p>
  <figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">x=[<span class="number">2</span>,<span class="number">2</span>];</span><br></pre></td></tr></table></figure>
</li>
<li><p>其次,构造出一个评价函数(或称目标函数).文件<code>OptFun.m</code>.</p>
  <figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">function</span> <span class="title">y</span>=<span class="title">OptFun</span><span class="params">(x)</span></span></span><br><span class="line">	y=x(<span class="number">1</span>)^<span class="number">2</span>+x(<span class="number">2</span>)^<span class="number">2</span>;</span><br><span class="line"><span class="keyword">end</span></span><br></pre></td></tr></table></figure>
</li>
<li><p>接着,构造一个能够不断<strong>根据旧解</strong>产生新解的函数.这里我们根据旧解以正态随机函数的形式产生新解.文件<code>Arrise.m</code>.</p>
  <figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">function</span> <span class="title">X</span>=<span class="title">Arrise</span><span class="params">(x)</span></span></span><br><span class="line">	X(<span class="number">1</span>)=normrnd(x(<span class="number">1</span>),<span class="number">2</span>);</span><br><span class="line">	X(<span class="number">2</span>)=normrnd(x(<span class="number">2</span>),<span class="number">2</span>);</span><br><span class="line"><span class="keyword">end</span></span><br></pre></td></tr></table></figure>
</li>
<li><p>复制<code>EzSA.m</code>到文件夹.</p>
</li>
<li><p>最后,调用现成的模拟退火函数EzSA.文件<code>main.m</code>.</p>
  <figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">[ x,res ]=EzSA(x,@myFirstSA,@Arrise)</span><br></pre></td></tr></table></figure>
<p>  如果你看到一个进度条,那么恭喜你,你已经会使用模拟退火算法了!</p>
<p>  让我们看看结果:</p>
<p>  <img src="/数模-模拟退火——史上最简单的智能算法/res1.jpg" alt="res1"></p>
<p>  图像记载了我们之前尝试的解值,可以看出在数次迭代后数值处于稳定状态,表示这次模拟退火算法成功了.</p>
<p>  同时,x 返回较优解,res返回较优值.</p>
  <figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">x =</span><br><span class="line">   <span class="number">-0.0035</span>   <span class="number">-0.0027</span></span><br><span class="line">res =</span><br><span class="line">   <span class="number">1.9562e-05</span></span><br></pre></td></tr></table></figure>
</li>
</ul>
<p><strong>让我们总结一下模拟退火函数的使用步骤:</strong></p>
<ol>
<li>提供或初始化一个初始解.</li>
<li>构造出一个评价函数(或称目标函数),该函数接收解,并返回一个数值(视值越小解越优).</li>
<li>构造一个能够不断<strong>根据旧解</strong>产生新解的函数(注意,这个函数的设计优劣直接影响到模拟退火效果的好坏).</li>
<li>调用现成的模拟退火函数EzSA(初始解,评价函数句柄,产生新解函数句柄).</li>
<li>一段时间后模拟退火算法结束,返回较优解和解值</li>
</ol>
<h2 id="例2-旅行商问题-TSP"><a href="#例2-旅行商问题-TSP" class="headerlink" title="例2:旅行商问题(TSP)"></a>例2:旅行商问题(TSP)</h2><p>现有五个城市,彼此间距离如图所示,现在旅行商需要经过所有城市一次并回到出发点.我们需要为他规划最短路线.</p>
<p><img src="/数模-模拟退火——史上最简单的智能算法/graph.jpg" alt="graph"></p>
<ul>
<li><p>首先,以邻接矩阵存储图并提供初始解.文件<code>main.m</code>.</p>
  <figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">global</span> n    <span class="comment">%n为城市数,由于在无法将n以参数形式传递给计算距离的函数,故声明为全局变量</span></span><br><span class="line"><span class="keyword">global</span> graph <span class="comment">%同上</span></span><br><span class="line">n=<span class="number">5</span>;</span><br><span class="line">graph=[<span class="number">0</span>,<span class="number">7</span>,<span class="number">6</span>,<span class="number">1</span>,<span class="number">3</span>;<span class="number">7</span>,<span class="number">0</span>,<span class="number">3</span>,<span class="number">7</span>,<span class="number">8</span>;<span class="number">6</span>,<span class="number">3</span>,<span class="number">0</span>,<span class="number">12</span>,<span class="number">11</span>;<span class="number">1</span>,<span class="number">7</span>,<span class="number">12</span>,<span class="number">0</span>,<span class="number">2</span>;<span class="number">3</span>,<span class="number">8</span>,<span class="number">11</span>,<span class="number">2</span>,<span class="number">0</span>];</span><br><span class="line"></span><br><span class="line">city=<span class="number">1</span>:<span class="number">5</span>; <span class="comment">%初始解</span></span><br></pre></td></tr></table></figure>
</li>
<li><p>其次是评价函数,设city为五个城市的访问顺序.文件<code>computerTour.m</code>.</p>
  <figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">function</span> <span class="title">len</span>=<span class="title">computerTour</span><span class="params">(city)</span>   %计算路线总长度，每个城市只计算和下家城市之间的距离。</span></span><br><span class="line">    <span class="keyword">global</span> n <span class="comment">%获取n为城市数</span></span><br><span class="line">    <span class="keyword">global</span> graph</span><br><span class="line">    len=<span class="number">0</span>;</span><br><span class="line">    <span class="keyword">for</span> <span class="built_in">i</span>=<span class="number">1</span>:n<span class="number">-1</span></span><br><span class="line">        len=len+graph(city(<span class="built_in">i</span>),city(<span class="built_in">i</span>+<span class="number">1</span>));</span><br><span class="line">    <span class="keyword">end</span></span><br><span class="line">    len=len+graph(city(n),city(<span class="number">1</span>));</span><br><span class="line"><span class="keyword">end</span></span><br></pre></td></tr></table></figure>
</li>
<li><p>接着,根据旧解产生新解的函数.文件<code>perturbTour.m</code>.</p>
  <figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">function</span> <span class="title">city</span>=<span class="title">perturbTour</span><span class="params">(city)</span></span></span><br><span class="line">    <span class="comment">%随机置换两个不同的城市的坐标</span></span><br><span class="line">    <span class="comment">%产生随机扰动</span></span><br><span class="line">    <span class="keyword">global</span> n</span><br><span class="line">    p1=randi([<span class="number">1</span>,n]);</span><br><span class="line">    p2=randi([<span class="number">1</span>,n]);</span><br><span class="line">    tmp=city(p1);</span><br><span class="line">    city(p1)=city(p2);</span><br><span class="line">    city(p2)=tmp;</span><br><span class="line"><span class="keyword">end</span></span><br></pre></td></tr></table></figure>
</li>
<li><p>最后,调用模拟退火函数(与第一步写在同一文件),并运行.文件<code>main.m</code></p>
  <figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">[city,res]=EzSA(city,@computerTour,@perturbTour)</span><br></pre></td></tr></table></figure>
</li>
</ul>
<p>结果:<br>    <figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">city =</span><br><span class="line">    <span class="number">4</span>     <span class="number">1</span>     <span class="number">3</span>     <span class="number">2</span>     <span class="number">5</span></span><br><span class="line">res =</span><br><span class="line">    <span class="number">20</span></span><br></pre></td></tr></table></figure></p>
<h1 id="原理讲解"><a href="#原理讲解" class="headerlink" title="原理讲解"></a>原理讲解</h1><p>源代码:<br><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">function</span> <span class="params">[X resEnd]</span>=<span class="title">EzSA</span><span class="params">(X,ObjFun,ArriseNew,iter,zero)</span></span></span><br><span class="line">    [ra,co]=<span class="built_in">size</span>(X);</span><br><span class="line">    RES=[ObjFun(X)];  <span class="comment">%每次迭代后的结果</span></span><br><span class="line">    temperature=<span class="number">100</span>*co;      <span class="comment">%初始温度</span></span><br><span class="line">    <span class="keyword">if</span> nargin==<span class="number">3</span></span><br><span class="line">        zero=<span class="number">1e-2</span>;</span><br><span class="line">        iter=<span class="number">5e2</span>;               <span class="comment">%内部蒙特卡洛循环迭代次数</span></span><br><span class="line">    <span class="keyword">end</span></span><br><span class="line">    <span class="keyword">if</span> nargin==<span class="number">4</span></span><br><span class="line">        zero=<span class="number">1e-2</span>;</span><br><span class="line">    <span class="keyword">end</span></span><br><span class="line"></span><br><span class="line">    h=waitbar(<span class="number">0</span>,<span class="string">'SAing....'</span>);</span><br><span class="line">    <span class="keyword">while</span> temperature&gt;zero    <span class="comment">%停止迭代温度</span></span><br><span class="line">        <span class="keyword">for</span> <span class="built_in">i</span>=<span class="number">1</span>:iter     <span class="comment">%多次迭代扰动，一种蒙特卡洛方法，温度降低之前多次实验</span></span><br><span class="line">            preRes=ObjFun(X);         <span class="comment">%目标函数计算结果</span></span><br><span class="line">            tmpX=ArriseNew(X);      <span class="comment">%产生随机扰动</span></span><br><span class="line">            newRes=ObjFun(tmpX);     <span class="comment">%计算新结果</span></span><br><span class="line"></span><br><span class="line">            delta_e=newRes-preRes;  <span class="comment">%新老结果的差值，相当于能量</span></span><br><span class="line">            <span class="keyword">if</span> delta_e&lt;<span class="number">0</span>        <span class="comment">%新结果好于旧结果，用新路线代替旧路线</span></span><br><span class="line">                X=tmpX;</span><br><span class="line">            <span class="keyword">else</span>                        <span class="comment">%温度越低，越不太可能接受新解；新老距离差值越大，越不太可能接受新解</span></span><br><span class="line">                <span class="keyword">if</span> <span class="built_in">exp</span>(-delta_e/temperature)&gt;<span class="built_in">rand</span>() <span class="comment">%以概率选择是否接受新解 p=exp(-ΔE/T)</span></span><br><span class="line">                    X=tmpX;      <span class="comment">%可能得到较差的解</span></span><br><span class="line">                <span class="keyword">end</span></span><br><span class="line">            <span class="keyword">end</span></span><br><span class="line">        <span class="keyword">end</span></span><br><span class="line">        RES=[RES ObjFun(X)];</span><br><span class="line">        temperature=temperature*<span class="number">0.99</span>;   <span class="comment">%温度不断下降</span></span><br><span class="line">        waitbar((<span class="built_in">log</span>(temperature/(<span class="number">100</span>*co))/<span class="built_in">log</span>(<span class="number">0.99</span>))/(<span class="built_in">log</span>(zero/(<span class="number">100</span>*co))/<span class="built_in">log</span>(<span class="number">0.99</span>)),h,sprintf(<span class="string">'Now Temperature:%.2f'</span>,temperature));</span><br><span class="line">    <span class="keyword">end</span></span><br><span class="line">    close(h)</span><br><span class="line">    <span class="built_in">plot</span>(RES);</span><br><span class="line">    resEnd=RES(<span class="keyword">end</span>);</span><br><span class="line"><span class="keyword">end</span></span><br></pre></td></tr></table></figure></p>
<p>结合代码再看开头的流程图.</p>
<p>初始化,计算初始解的解值,设置初始温度.<br>模拟退火结构上就是两重循环,外部循环检查温度并降温,内部不断地产生新解并与旧解比较.</p>
<p>若新解优于旧解则新解无条件被旧解替代.<br>否则,有一定概率(<code>exp(-ΔE/T)</code>)新解取代旧解.<strong>注意这个环节正是模拟退火能跳脱局部最优解,取得全局最优解的关键</strong>.</p>
<p>由此,我们可以得知影响模拟退火效果的主要因素有:</p>
<ul>
<li>终止温度.一般上,终止温度越低,取得解越优.</li>
<li>内部迭代次数.一般上,内部迭代次数越多,取得解越优.</li>
<li>产生新解函数.</li>
</ul>
<h1 id="总结"><a href="#总结" class="headerlink" title="总结"></a>总结</h1><p>模拟退火是对热力学退火过程的模拟,使算法在多项式时间内能给出一个近似最优解.由于MATLAB自带的模拟退火工具箱调用复杂且执行效果不理想,本文给出了较简单的函数原型和调用方法.该算法也包含以下优缺点(个人见解):</p>
<p>优点:</p>
<ul>
<li>相较于一般的蒙特卡洛算法,有更少的尝试次数,同时实现上并不比蒙特卡洛花更多时间.</li>
<li>相较于遗传算法等大型智能算法,模拟退火实现简单,并能返回较满意的结果.</li>
<li>目标函数可以自己定制,相较于普通的规划解析器,模拟退火能适用于更广的范围(NPC问题,甚至给给神经网络做优化).</li>
<li>对于离散型的变量有更优秀的效果.</li>
</ul>
<p>缺点:</p>
<ul>
<li>内部本质上还是蒙特卡洛算法,新解与旧解本质上无关联.</li>
<li>相较于遗传算法,模拟退火难以控制算法的运行时间,EzSA的后面两个可选参数就是内部迭代次数和0度温度.而迭代次数给少了效果不理想,给多了有会增加等待时间.</li>
<li>对连续型的规划问题效果并不好.</li>
</ul>
<p>样例和函数原型下载:<br><a href="https://github.com/Anemone95/matlab-sa" target="_blank" rel="noopener">https://github.com/Anemone95/matlab-sa</a></p>

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